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**Purpose**: To develop a fast landmark-based deformable registration method to capture the soft tissue transformation between the planning 3D CT images and treatment 3D cone-beam CT (CBCT) images for the adaptive external beam radiotherapy (EBRT).
**Method and Materials**: The developed method was based on a global-to-local landmark-based deformable registration algorithm. The landmarks were first acquired by applying a fast segmentation method using the active shape model. The global registration method was applied to establish a registration framework. The Laplacian surface deformation (LSD) and Laplacian surface optimization (LSO) method were then employed for local deformation and remeshing respectively to reach an optimal registration solution. In LSD, the deformed mesh is generated by minimizing the quadratic energy to keep the shape and to move control points to the target position. In LSO, a mesh is reconstructed by minimizing the quadratic energy to smooth the object by minimizing the difference while keeping the landmarks unchanged. The method was applied on 8 EBRT prostate datasets. The distance and volume based estimators were used to evaluate the results. The target volumes delineated by physicians were used as gold standards in the evaluation.
**Results**: The entire segmentation and registration processing time was within 1 minute for all the datasets. The mean distance estimators ranged from 0.43 mm to 2.23 mm for the corresponding model points between the treatment CBCT images and the registered planning images. The mean overlap ratio ranged from 85.5% to 93.2% of the prostate volumes after registration. These results demonstrated reasonably good agreement between the developed method and the gold standards.
**Conclusion**: A novel and fast landmark-based deformable registration method is developed to capture the soft tissue transformation between the planning and treatment images for prostate target volumes. The results show that with the method the image registration and transformation can be completed within one minute and has the potential to be applied to real-time adaptive radiotherapy.

This Prostate cancer is the most commonly diagnosed cancer in men in the United States and is the second leading cause of cancer related deaths in males [

Image guided radiotherapy (IGRT) has been developed to improve the detection of target deviations relative to the target position in an approved radiotherapy treatment plan [

Various imaging based rigid registration methods have been developed to automate the process of determining patient positioning [7-12]. These methods have been aimed at registering 3D CT data to 2D portal images. The significant downfall of these methods is that only rigid pelvic bony landmarks are used while the location of prostate itself is not considered. In one study [

Various full 3D deformable registration methods have been developed to register images in the prostate region, such as biomechanical models, conformal mapping, thin plate spline, diffusion based models, and free form deformation [14-26].

Biomechanical models can be time consuming since the mechanical information of the tissue is required to perform the deformation [14-16]. Conformal mapping method is applied to map bladder deformations and difficulties can be encountered to model the interfaces between structures [

In one of our previous studies [

To overcome the limitations of the previous method, we developed a new global-to-local deformable registration method which incorporated both computationally efficient landmark points based global registration method and a new computationally efficient local deformable mesh model based local registration algorithm.

In the global registration, the corresponding landmark points were first acquired by applying the anatomy-constrained robust active shape models (ACRASM) segmentation method [

In the local registration, the Laplacian surface deformation (LSD) [

The flow chart of the segmentation and registration process is shown in

Eight different patients undergoing external beam prostate

radiotherapy were randomly selected for the study and evaluations. Each of the patients had one set of planning CT images and one set of CBCT images. The planning CT (pCT) images were acquired with an AcQSim CT scanner (Philips Medical Systems, Andover, MA). The number of slices was different for different patients, ranging from 85 slices to 121 slices. Image slice was 512 × 512 pixels with a voxel dimension of 0.94 × 0.94 × 3 mm^{3}. The treatment CBCT images were acquired by an On-Board Imager (OBI, Varian Medical Systems). The number of slices was also different for different patients, ranging from 55 slices to 61 slices. Each slice had 512 × 512 pixels with a voxel dimension of 0.3 × 0.3 × 2.5 mm^{3}.

The target volumes in pCT images were contoured for treatment planning purposes on an Eclipse treatment planning system (Varian Medical Systems, CA) by an experienced radiation oncologist specializing in prostate cancer management. The target volumes were also manually delineated by the same radiation oncologist on the CBCT images. These manually delineated target volumes from the CBCT images were used as the benchmarks for comparison and evaluation. All the contours were rviewed and confirmed by another radiation oncologist.

Since the pCT and CBCT image slice had different voxel dimensions and different field-of-view size, all the image datasets were resampled in both pCT and CBCT to a uniform voxel dimension 0.3 × 0.3 × 0.3 mm^{3}. After the resampling process, the pCT and CBCT will have same voxel dimensions and similar field-of-view size.

The image intensity was first normalized to [0,1] and the histogram equalization method was applied to enhance the contrast of images in the pCT and CBCT before the image segmentation and deformable registration computation. This image contrast enhancement process helped to improve the accuracies of both the segmentation and the deformable registration.

As a second preprocessing step of the segmentation and deformable registration algorithm, were all automatically aligned in a common reference frame, using a rigid transformation based on the pelvic bony structures. These bony structures normally show less variability compared to soft tissues, and can be automatically identified.

All uniformly resampled image datasets will be first cropped into the volume of interest shown in

The detailed description of the method of segmentation was presented in a previous study [

Enhanced planning CT images unfolded from a single 3D planning CT prostate volume were segmented using the anatomy-constrained robust active shape models (ACRASM) [

Enhanced CBCT images unfolded from a single 3D CBCT prostate volume were also segmented using ACRASM method, the same segmentation method by which the enhanced planning CT images were segmented.

The upper, bottom, left and right four points were also selected as landmark points of model S1 and the rest points of model S1 were regular points in each image of the 3D CBCT prostate volume. The landmark points of model S1 in the 3D CBCT prostate volume would be used in global registration as well as the local registration.

A global-to-local landmark points based deformable registration framework was utilized. We denote the 3D mesh in the planning CT images as the source shape and the 3D mesh in the CBCT treatment images as the target shape.

Global registration was developed by applying Procrustes analysis to approximately find similarity transformation, such as the translation, scaling and rotation matrix, based on the corresponding landmark points from the planning CT images and CBCT images. Let P = {p^{1}, p^{2}, , p^{n}} and Q = {q^{1}, q^{2}, , q^{n}} be two sets of corresponding landmark points from the planning CT images and CBCT images respectively. The goal is to find an optimal similarity transformation based on the two sets of corresponding landmark points by minimizing the following the least square function:

where, T, S, R are translation matrix, scaling matrix and rotation matrix respectively.

After solving this nine dimensional similarity search problem, the entire model can be transformed using these transformation matrices and roughly registered with the other one. After the global deformation and registration, non-rigid local deformable registration using Laplacian surface deformation optimization was employed to achieve optimal accuracy.

2.4.2.1. Local deformation using Laplacian Surface Deformation (LSD)

Laplacian surface is defined on differential coordinates. It represents each vertex point of a mesh as the difference between the point and its neighborhoods. The inputs of LSD are two sets of the control points in the planning CT and CBCT images and the initial mesh of the source shape. The mesh vertices are the vertex points which are divided into two parts. One part is control points or landmark points and the other part is regular points. In the Laplacian surface deformation process, the control points in the source images are moved to target images directly and the deformation of the regular points are calculated by LSD. Note that after global deformation, the displacements of control points are restricted in a local range. The output of LSD is the deformed mesh of the source shape.

Let the initial source mesh MS be described by a pair (VS, ES), where VS ={v^{1}, v^{2}, , v^{n}} describes the geometric positions of the vertex points in R3 and ES describes the connectivity. The neighborhood ring of a vertex point i is the set of adjacent vertex points and the degree d_{i} of this vertex point is the number of elements in N_{S}_{,i}. Instead of using absolute coordinates VS, the mesh geometry is described as a set of differentials Δ =, is the coordinate i, which will be represented by the difference between v_{i} and the average of its neighbors:

The deformed mesh can be generated by minimizing the quadratic energy function:

where, ' is Laplacian coordinate after deformation and v_{i}' is Cartesian coordinate after deformation, C is the set of control points, Ti is transformation matrix for each vertex point i. The first half is to keep the similarity of the current shape and the shape generated in the previous time step and the second half try to move control points to target position. We chosed n equals 420 to present the prostate mesh model and 84 points were selected as the control points which was explained in Section 2.3.

Ti needs to be well constrained to avoid a membrane solution, which shows losing all geometric details. Because of this T_{i} should include rotation, isotropic scales and translations, we find anisotropic scales should not be allowed, as they will allow removing the normal component from Laplacian coordinates. Specifically, T_{i} is defined as:

This matrix is a good linear approximation for rotations with small angles. Furthermore, it only allows isotropic scales by enforcing the same values for diagonal elements.

The quadratic energy function can be minimized iteratively by finding s, h and t for T_{i} and applying the transformation on each vertex coordinates. We use the changing of residuals as the convergence criterion. When its value is smaller than a threshold (e.g., 1E - 6), we assume that the system converges and this minimization problem is solved. Note that the transformation Ti is an approximation of the isotropic scaling and rotations when the rotation angle is small. In this paper, the major rotation has been handled in the global deformation so that the local rotation fits the small angle assumption well.

2.4.2.2. Local Deformation Using Laplacian Surface Optimization (LSO)

LSO is used to improve triangle quality of a surface mesh. The inputs are landmark points and the initial surface mesh and the output is an optimized surface mesh. We will use the same notation as in Section 2.5.2.1.

Assume V is the matrix representation of VS. The transformation between vertex coordinates V and Laplacian coordinates Δ can be described in matrix algebra. Let N be the mesh adjacency matrix and D = diag(d_{1}, ···, d_{n}) be the degree matrix. Then, , where for the uniform weights.

Using a subset of m landmark points, a mesh can be reconstructed from connectivity information alone. Here the selection of A is the same as the control points selected in II.E.2.1. Positions of the reconstructed object can be solved separately by minimizing the quadratic energy:

where V_{p}' is the vertex coordinates of the reconstructed object, v_{ap} are landmark points. The first half is Laplacian constrains to smooth the object by minimizing the difference, and the second half is positional constrains to keep landmark points unchanged. In practice, with m landmark points, the overdetermine linear system:

is solved in the least squares method using the method of normal equations. The first n rows of are the Laplacian constrains, corresponding to the first part, while the next m rows are the positional constrains, corresponding to the second part .

I_{ap} is the index matrix of v_{ap}, which maps each v'_{ap} to v_{ap}. The reconstructed shape is generally smooth, with the possible exception of small areas around landmark points. The minimization procedure moves each vertex point to the centroid of its ring, since the uniform Laplacian L is used, resulting in good inner fairness. The main computation cost of this algorithm is big matrix multiplication and inverse. Since A is sparse matrix, ATA is sparse symmetric definite matrix. The conjugate gradient algorithm can be employed to solve the system.

The quantitative validation of the proposed deformable registration algorithm for prostate cancer image volumes was designed to include three components: distancebased estimators, volume-based estimators and registration efficiency. For all three evaluations, the proposed method was compared against the corresponding benchmarks as well as the registration results obtained using our previous deformable mesh model based image Registration method.

The Distance-Based Estimators and the Volume-Based EstimatorsThe distance-based estimators include the mean distance and the root mean square error (RMSE). The mean distance, the root-mean-square error (RMSE) and the max distance are calculated the corresponding prostate points between the treatment CBCT images and the registered planning images. The corresponding prostate points include two parts of points, corresponding landmark points and corresponding non-landmark points. The corresponding landmark points was acquired for the global registration step in the Sections 2.3. and 2.4. The corresponding non-landmark points which are defined as the closest point to the original mesh point.

The volume-based estimators include the volumetric true positive (TP), volumetric false negative (FN), volumetric the false positive (FP) and Dice similarity coefficient

between the results of proposed deformable registration method and benchmarks. We used this similarity measure since it accurately reflected the overall volumetric overlapping between binary objects and also included the information about the major causes of the differences, such as overestimate or underestimate.

The quantitative results are summarized in